Determining if series converges or diverges

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In summary, the conversation discusses the validity of using the limit comparison test to determine the convergence or divergence of a series and the relationship between the terms in the series. The use of the ratio test is also mentioned as a possible alternative.
  • #1
Sunwoo Bae
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Homework Statement
Determine if the following series converges or diverges using any appropriate tests
Relevant Equations
limit comparison test
1633162415960.png

Is it valid to use limit comparison test to compute the following series?
If it is, would my reasoning be valid?

Thank you!
 
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  • #2
What precisely is your reasoning?
 
  • #3
An orthodox way is to investigate
[tex]|\frac{a_{n+1}}{a_n}|[/tex]
is greater or less than 1.
 
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  • #4
Sunwoo Bae said:
Homework Statement:: Determine if the following series converges or diverges using any appropriate tests
Relevant Equations:: limit comparison test

View attachment 290042
Is it valid to use limit comparison test to compute the following series?
If it is, would my reasoning be valid?

Thank you!
What does the limit comparison test say exactly?
Isn't there a specific relation to be fulfilled between ##a_n## and ##b_n##?
Do your successions ##a## and ##b## fulfill such relation?
 
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  • #5
Sunwoo Bae said:
Homework Statement:: Determine if the following series converges or diverges using any appropriate tests
Relevant Equations:: limit comparison test

View attachment 290042
Is it valid to use limit comparison test to compute the following series?
If it is, would my reasoning be valid?

Thank you!

Is [tex]
\frac{n^n}{n!} = \frac{n}{1} \frac{n}{2} \cdots \frac{n}{n-1} \frac{n}{n}
[/tex] greater than, or less than, 1 for large [itex]n[/itex]? Given that, which of the following is true:
1. [itex]a_n < b_n[/itex] for large [itex]n[/itex].
2. [itex]a_n > b_n[/itex] for large [itex]n[/itex].

How does that affect the comparison test?
 
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  • #6
I personally think the ratio test would be the easiest to use, here.
 

FAQ: Determining if series converges or diverges

1. What is the definition of convergence and divergence in a series?

Convergence in a series means that the terms of the series approach a finite limit as the number of terms increases. Divergence means that the terms of the series do not approach a finite limit and the series does not have a sum.

2. How do you determine if a series converges or diverges?

There are several tests that can be used to determine convergence or divergence, such as the ratio test, the root test, and the comparison test. These tests compare the given series to a known series with a known convergence or divergence.

3. What is the difference between absolute and conditional convergence?

Absolute convergence means that the series converges when the absolute value of the terms is taken. Conditional convergence means that the series converges, but not when the absolute value of the terms is taken.

4. Can a series converge to more than one value?

No, a series can only have one sum. If a series converges, it will always have the same sum regardless of the order in which the terms are added.

5. What is the significance of a series converging or diverging?

The convergence or divergence of a series is important in determining the behavior and properties of the series. It also has applications in various fields of mathematics and science, such as in calculus, physics, and engineering.

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